Calculus Examples

$y = x y^{2} + 4 x - y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x y^{2} + 4 x - y)$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $x y^{2} + 4 x - y$ with respect to $x$ is $\frac{d}{d x} [x y^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- y]$ .

$\frac{d}{d x} [x y^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- y]$

Step 3.2

Evaluate $\frac{d}{d x} [x y^{2}]$ .

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Step 3.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y^{2}$ .

$x \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- y]$

Step 3.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.2.2.1

To apply the Chain Rule, set $u$ as $y$ .

$x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- y]$

Step 3.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$x (2 u \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- y]$

Step 3.2.2.3

Replace all occurrences of $u$ with $y$ .

$x (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- y]$

Step 3.2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$x (2 y y') + y^{2} \frac{d}{d x} [x] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- y]$

Step 3.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$x (2 y y') + y^{2} \cdot 1 + \frac{d}{d x} [4 x] + \frac{d}{d x} [- y]$

Step 3.2.5

Move $2$ to the left of $x$ .

$2 \cdot x y y' + y^{2} \cdot 1 + \frac{d}{d x} [4 x] + \frac{d}{d x} [- y]$

Step 3.2.6

Multiply $y^{2}$ by $1$ .

$2 x y y' + y^{2} + \frac{d}{d x} [4 x] + \frac{d}{d x} [- y]$

Step 3.3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$2 x y y' + y^{2} + 4 \frac{d}{d x} [x] + \frac{d}{d x} [- y]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 x y y' + y^{2} + 4 \cdot 1 + \frac{d}{d x} [- y]$

Step 3.3.3

Multiply $4$ by $1$ .

$2 x y y' + y^{2} + 4 + \frac{d}{d x} [- y]$

Step 3.4

Evaluate $\frac{d}{d x} [- y]$ .

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Step 3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- y$ with respect to $x$ is $- \frac{d}{d x} [y]$ .

$2 x y y' + y^{2} + 4 - \frac{d}{d x} [y]$

Step 3.4.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$2 x y y' + y^{2} + 4 - y'$

Step 3.5

Reorder terms.

$y^{2} + 2 x y y' + 4 - y'$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = y^{2} + 2 x y y' + 4 - y'$

Step 5

Solve for $y'$ .

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Step 5.1

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$y^{2} + 2 x y y' + 4 - y' = y'$

Step 5.2

Move all terms containing $y'$ to the left side of the equation.

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Step 5.2.1

Subtract $y'$ from both sides of the equation.

$y^{2} + 2 x y y' + 4 - y' - y' = 0$

Step 5.2.2

Subtract $y'$ from $- y'$ .

$y^{2} + 2 x y y' + 4 - 2 y' = 0$

Step 5.3

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.3.1

Subtract $y^{2}$ from both sides of the equation.

$2 x y y' + 4 - 2 y' = - y^{2}$

Step 5.3.2

Subtract $4$ from both sides of the equation.

$2 x y y' - 2 y' = - y^{2} - 4$

Step 5.4

Factor $2 y'$ out of $2 x y y' - 2 y'$ .

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Step 5.4.1

Factor $2 y'$ out of $2 x y y'$ .

$2 y' (x y) - 2 y' = - y^{2} - 4$

Step 5.4.2

Factor $2 y'$ out of $- 2 y'$ .

$2 y' (x y) + 2 y' \cdot - 1 = - y^{2} - 4$

Step 5.4.3

Factor $2 y'$ out of $2 y' (x y) + 2 y' \cdot - 1$ .

$2 y' (x y - 1) = - y^{2} - 4$

Step 5.5

Divide each term in $2 y' (x y - 1) = - y^{2} - 4$ by $2 (x y - 1)$ and simplify.

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Step 5.5.1

Divide each term in $2 y' (x y - 1) = - y^{2} - 4$ by $2 (x y - 1)$ .

$\frac{2 y' (x y - 1)}{2 (x y - 1)} = \frac{- y^{2}}{2 (x y - 1)} + \frac{- 4}{2 (x y - 1)}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $2$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{2 y' (x y - 1)}{2 (x y - 1)} = \frac{- y^{2}}{2 (x y - 1)} + \frac{- 4}{2 (x y - 1)}$

Step 5.5.2.1.2

Rewrite the expression.

$\frac{y' (x y - 1)}{x y - 1} = \frac{- y^{2}}{2 (x y - 1)} + \frac{- 4}{2 (x y - 1)}$

Step 5.5.2.2

Cancel the common factor of $x y - 1$ .

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Step 5.5.2.2.1

Cancel the common factor.

$\frac{y' (x y - 1)}{x y - 1} = \frac{- y^{2}}{2 (x y - 1)} + \frac{- 4}{2 (x y - 1)}$

Step 5.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- y^{2}}{2 (x y - 1)} + \frac{- 4}{2 (x y - 1)}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Move the negative in front of the fraction.

$y' = - \frac{y^{2}}{2 (x y - 1)} + \frac{- 4}{2 (x y - 1)}$

Step 5.5.3.1.2

Cancel the common factor of $- 4$ and $2$ .

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Step 5.5.3.1.2.1

Factor $2$ out of $- 4$ .

$y' = - \frac{y^{2}}{2 (x y - 1)} + \frac{2 \cdot - 2}{2 (x y - 1)}$

Step 5.5.3.1.2.2

Cancel the common factors.

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Step 5.5.3.1.2.2.1

Cancel the common factor.

$y' = - \frac{y^{2}}{2 (x y - 1)} + \frac{2 \cdot - 2}{2 (x y - 1)}$

Step 5.5.3.1.2.2.2

Rewrite the expression.

$y' = - \frac{y^{2}}{2 (x y - 1)} + \frac{- 2}{x y - 1}$

Step 5.5.3.1.3

Move the negative in front of the fraction.

$y' = - \frac{y^{2}}{2 (x y - 1)} - \frac{2}{x y - 1}$

Step 5.5.3.2

To write $- \frac{2}{x y - 1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y' = - \frac{y^{2}}{2 (x y - 1)} - \frac{2}{x y - 1} \cdot \frac{2}{2}$

Step 5.5.3.3

Write each expression with a common denominator of $2 (x y - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.3.1

Multiply $\frac{2}{x y - 1}$ by $\frac{2}{2}$ .

$y' = - \frac{y^{2}}{2 (x y - 1)} - \frac{2 \cdot 2}{(x y - 1) \cdot 2}$

Step 5.5.3.3.2

Reorder the factors of $(x y - 1) \cdot 2$ .

$y' = - \frac{y^{2}}{2 (x y - 1)} - \frac{2 \cdot 2}{2 (x y - 1)}$

Step 5.5.3.4

Combine the numerators over the common denominator.

$y' = \frac{- y^{2} - 2 \cdot 2}{2 (x y - 1)}$

Step 5.5.3.5

Multiply $- 2$ by $2$ .

$y' = \frac{- y^{2} - 4}{2 (x y - 1)}$

Step 5.5.3.6

Factor $- 1$ out of $- y^{2}$ .

$y' = \frac{- (y^{2}) - 4}{2 (x y - 1)}$

Step 5.5.3.7

Rewrite $- 4$ as $- 1 (4)$ .

$y' = \frac{- (y^{2}) - 1 (4)}{2 (x y - 1)}$

Step 5.5.3.8

Factor $- 1$ out of $- (y^{2}) - 1 (4)$ .

$y' = \frac{- (y^{2} + 4)}{2 (x y - 1)}$

Step 5.5.3.9

Simplify the expression.

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Step 5.5.3.9.1

Rewrite $- (y^{2} + 4)$ as $- 1 (y^{2} + 4)$ .

$y' = \frac{- 1 (y^{2} + 4)}{2 (x y - 1)}$

Step 5.5.3.9.2

Move the negative in front of the fraction.

$y' = - \frac{y^{2} + 4}{2 (x y - 1)}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{y^{2} + 4}{2 (x y - 1)}$